A SECOND COURSE IN COMPLEX ANALYSIS.

By William A. Veech ’60.New York: W. A. Benjamin, Inc., 1967.246 pp. $8.75.

The "New Math" phenomenon is not restricted to the elementary school scene. Here is a contemporary treatment of an ancient and honorable subject normally riven in the upper divisions of the college. As the title suggests, this book is intended as a text in a second course in complex analysis. The author assumes a standard introductory course in complex analysis as prerequisite, and proceeds to conduct a guided tour through some of the major landmarks of the field.

Because complex analysis is such an enormous field, containing some of the best work of two centuries of mathematics, such a course is always given "at the pleasure of the instructor," and reflects his interests, taste, and style. Mr. Veech has chosen carefully from endless possibilities a half-dozen subjects of particular interest to modern research and has presented them in the language and style of the current journals. He begins with a careful discussion of analytic functions, their germs and their continuations; then turns to geometric considerations, including linear fractional transformations, non-Euclidean geometries, and the mapping theorems; and then discusses the properties of modular functions. In the final chapters he takes up the study of representations by infinite products, and concludes with an elegant proof of the celebrated prime number theorem.

The result is a very attractive, smooth, and informative book, which covers a lot of hallowed ground and ought to be great fun to teach from. Another author might have chosen a different assortment of subjects, or placed his emphasis in a different way, but he would find it hard to improve on the standard set by Mr. Veech.

Associate Professor of Mathematics at Dartmouth College, Mr. Prosser teaches an advanced graduate course in Applied Mathematics.